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arXiv · 1907.06025

Convex hypersurface theory in contact topology

Joseph Breen·Austin Christian·Ko Honda·Yang Huang
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Abstract

We lay the foundations of convex hypersurface theory in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^0$-approximated by a convex one. We also prove that a $C^0$-generic family of mutually disjoint closed hypersurfaces parametrized by $t\in[0,1]$ is convex except at finitely many times $t_1,\dots,t_N$, and that crossing each $t_i$ corresponds to a bypass attachment. As an application, we prove the existence of compatible (relative) open book decompositions for contact manifolds.

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Joseph Breen, Austin Christian, Ko Honda, Yang Huang. 2019-07-13. Convex hypersurface theory in contact topology. https://arxiv.org/abs/1907.06025

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